Peacock patterns and resurgence in complex Chern-Simons theory

Garoufalidis, Stavros1,2; Gu, Jie3; Marino, Marcos4

2023-09-01

10.1007/s40687-023-00391-1

RESEARCH IN THE MATHEMATICAL SCIENCES   影响因子和分区

2522-0144

2197-9847

The partition function of complex Chern-Simons theory on a 3-manifold with torus boundary reduces to a finite-dimensional state-integral which is a holomorphic function of a complexified Planck's constant t in the complex cut plane and an entire function of a complex parameter u. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear q-difference equation. We further conjecture that a distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte-Gaiotto-Gukov. We provide proofs of our statements regarding the q-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic 4(1) and 5(2) knots.

Chern-Simons theory Holomorphic blocks State-integrals Knots 3-manifolds Resurgence Perturbative series Borel resummation Stokes automorphisms Stokes constants q-holonomic modules q-difference equations DT-invariants BPS states Peacocks Meromorphic quantum Jacobi forms

SCI

英语

第一 ; 通讯

NCCR "The Mathematics of Physics" (SwissMAP)[51NF40-182902] ; ERC-SyG project "Recursive and Exact New Quantum Theory" (ReNewQuantum) from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program[810573]

Mathematics

Mathematics

WOS:001015068400001

SPRINGER INT PUBL AG

Web of Science

1.Southern Univ Sci & Technol, Int Ctr Math, Dept Math, Shenzhen, Peoples R China
2.Max Planck Inst Math, D-53111 Bonn, Germany
3.Univ Geneva, Dept Phys Theor, CH-1211 Geneva, Switzerland
4.Univ Geneva, Dept Phys Theor, Sect Math, CH-1211 Geneva, Switzerland

Garoufalidis, Stavros,Gu, Jie,Marino, Marcos. Peacock patterns and resurgence in complex Chern-Simons theory[J]. RESEARCH IN THE MATHEMATICAL SCIENCES,2023,10(3).

Garoufalidis, Stavros,Gu, Jie,&Marino, Marcos.(2023).Peacock patterns and resurgence in complex Chern-Simons theory.RESEARCH IN THE MATHEMATICAL SCIENCES,10(3).

Garoufalidis, Stavros,et al."Peacock patterns and resurgence in complex Chern-Simons theory".RESEARCH IN THE MATHEMATICAL SCIENCES 10.3(2023).